Suppose we have a group of officers in six regiments, each regiment consisting of the same six ranks (say, a colonel, a lieutenant colonel, a major, a captain, a first lieutenant, and a second lieutenant). Is it possible to arrange these 36 officers into a 6 × 6 square so that no rank or regiment is repeated in any row or column? That is, each row and column must contain an officer of each regiment and of each rank.

In 1782 Leonhard Euler wrote, “After we have put a lot of thought into finding a solution, we have to admit that such an arrangement is impossible, though we can’t give a rigorous demonstration of this.” He saw that the equivalent problem is impossible in a 2 × 2 square and surmised that it’s impossible in every case where the side of the square contains 4k + 2 cells.

It wasn’t until 1901 that French mathematician Gaston Terry proved that the 6 × 6 square has no solution, and it wasn’t until 1960 that Euler’s conjecture about the pattern of impossible squares was proven wrong: In fact, the task is impossible only in these two cases, 2 × 2 and 6 × 6.

This is neat: Figure 1 in this physics paper is a 1:1 scale image of a primordial black hole whose mass is 5 times that of Earth.

A black hole of 10 Earth masses “is roughly the size of a ten pin bowling ball.”

(Jakub Scholtz and James Unwin, “What If Planet 9 Is a Primordial Black Hole?”, arXiv preprint arXiv:1909.11090 [2019].)

(Via Cliff Pickover’s excellent Twitter feed.)

In 1995 UCSD psychologist Diana Deutsch was fine-tuning the spoken commentary on a CD when she noticed something odd. When the phrase “sometimes behave so strangely” was repeated on a loop, it came to sound as though it were being sung rather than spoken. When the full surrounding passage was then played in its entirety, this phrase still sounded as though it were being sung (you can hear this here).

The phenomenon is not completely understood, but “the present experiments show that for a phrase to be heard as spoken or as sung, it does not need to have a set of physical properties that are unique to speech, or a different set of physical properties that are unique to song,” the researchers write. “Rather, we must conclude that, assuming the neural circuitries underlying speech and song are at some point distinct and separate, they can accept the same input, but process the information in different ways so as to produce different outputs.”

(Via MetaFilter.)

Humans aren’t the only species that tend to move to a musical beat: Animals that mimic vocally (such as parrots and, here, a sulphur-crested cockatoo) bob their heads and move their feet. Animals that don’t mimic vocally don’t do this. So possibly our urge to move to music is a by-product of our tendency to mimic vocally — it’s a motor response to something we hear.

(Aniruddh D. Patel, et al., “Experimental Evidence for Synchronization to a Musical Beat in a Nonhuman Animal,” Current Biology 19:10 [2009], 827-830; Adena Schachner, et al., “Spontaneous Motor Entrainment to Music in Multiple Vocal Mimicking Species,” Current Biology 19:10 [2009], 831-836.)

                                       1 
                                      1 1 
                                     1 0 1 
                                    1 0 0 1 
                                   1 0 0 0 1 
                                  1 0 0 0 0 1 
                                 1 0 0 0 0 0 1 
                                1 0 0 0 0 0 0 1 
                               1 0 0 0 0 0 0 0 1 
                              1 0 0 0 0 0 0 0 0 1 
                             1 0 0 0 0 0 0 0 0 0 1 
                            1 0 0 0 0 0 0 0 0 0 0 1 
                           1 0 0 0 0 0 0 0 0 0 0 0 1 
1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
  1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
   1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
    1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
     1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
      1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
       1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
        1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
         1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
          1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
           1 0 0 0 0 0 0 0 0 0 0 1 1 1 9 1 1 1 0 0 0 0 0 0 0 0 0 0 1 
            1 0 0 0 0 0 0 0 0 0 0 1 9 9 9 9 1 0 0 0 0 0 0 0 0 0 0 1 
             1 0 0 0 0 0 0 0 0 0 0 1 9 2 9 1 0 0 0 0 0 0 0 0 0 0 1 
            1 0 0 0 0 0 0 0 0 0 0 1 9 9 9 9 1 0 0 0 0 0 0 0 0 0 0 1 
           1 0 0 0 0 0 0 0 0 0 0 1 1 1 9 1 1 1 0 0 0 0 0 0 0 0 0 0 1 
          1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
         1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1  
        1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
       1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
      1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
     1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
    1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
   1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
  1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 
1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 
                           1 0 0 0 0 0 0 0 0 0 0 0 1 
                            1 0 0 0 0 0 0 0 0 0 0 1 
                             1 0 0 0 0 0 0 0 0 0 1 
                              1 0 0 0 0 0 0 0 0 1 
                               1 0 0 0 0 0 0 0 1 
                                1 0 0 0 0 0 0 1 
                                 1 0 0 0 0 0 1 
                                  1 0 0 0 0 1 
                                   1 0 0 0 1 
                                    1 0 0 1 
                                     1 0 1 
                                      1 1 
                                       1
 

Amazingly, this is a prime number. When the digits are assembled into one long string, beginning at the top of the star and reading each row left to right, they form a 1093-digit number whose only factors are 1 and itself.

It was discovered by Australian mathematician Michael Hartley. See more of his prime curios.

In 1894 Francis Galton experimented with conducting addition and subtraction by smell. He designed an apparatus that would produce whiffs of scented air and then memorized their combinations: “I taught myself to associate two whiffs of peppermint with one whiff of camphor; three of peppermint with one of carbolic acid, and so on.”

After practicing sums using the scents themselves, he moved on to doing them entirely in his imagination. “There was not the slightest difficulty in banishing all visual and auditory images from the mind, leaving nothing in the consciousness besides real or imaginary scents. In this way, without, it is true, becoming very apt at the process, I convinced myself of the possibility of doing sums in simple addition with considerable speed and accuracy solely by means of imaginary scents.”

He had similar success with subtraction, but didn’t try multiplication. And some further experiments seemed to show that “arithmetic by taste was as feasible as arithmetic by smell.”

(Francis Galton, “Arithmetic by Smell,” Psychological Review 1:1 [January 1894], 61-62.)

If a tetrahedron is constructed on a base with side lengths 125, 244, 267, then the remaining sides can take the shapes of three right triangles: (44, 240, 244), (44, 117, 125), (117, 240, 267).

And now the sum of the squares of the lengths of each pair of opposite sides in the tetrahedron is the same:

117² + 244² = 125² + 240² = 44² + 267² = 73225

(From Edward Barbeau, Power Play, 1997.)